There are probably a fair few chemicals that can dissolve rubber, but the best one I know of is acetone. You'd need to soak the rubber in plenty of acetone, rather than simply applying a little amount, which would quickly dry out.
A word of warning
Just a quick note before you get started. Acetone is highly flammable so you need to be really careful when using it. It can also be irritating to skin so don't get covered in the stuff!
Other uses for acetone
- Acetone is primarily used as a solvent, and often used to remove resin, glue, and grease.
- Acetone is also used in laboratories as a solvent in organic reactions.
- Many cosmetic and medical applications contain acetone, including acne treatments as a way to peel dry skin.
- As a domestic product, acetone is the primary ingredient in most nail polish removers, and is used to remove residue from glass and porcelain.
Here's a little video showing what acetone can do to Styrofoam: